190. Those who have not learnt how to take the [47]Altitude of any Celestial Phenomenon by a common Quadrant, nor know any thing of Plain Trigonometry, may pass over the first Article of this short Chapter, and take the Astronomer’s word for it, that the distances of the Sun and Planets are as stated in the first Chapter of this Book. But, to every one who knows how to take the Altitude of the Sun, the Moon, or a Star, and can solve a plain right-angled Triangle, the following method of finding the distances of the Sun and Moon will be easily understood.
Let BAG be one half of the Earth, AC it’s semi-diameter, S the Sun, m the Moon, and EKOL a quarter of the Circle described by the Moon in revolving from the Meridian to the Meridian again. Let CRS be the rational Horizon of an observer at A, extended to the Sun in the Heavens, and HAO his sensible Horizon; extended to the Moon’s Orbit. ALC is the Angle under which the Earth’s semi-diameter AC is seen from the Moon at L, which is equal to the Angle OAL, because the right lines AO and CL which include both these Angles are parallel. ASC is the Angle under which the Earth’s semi-diameter AC is seen from the Sun at S, and is equal to the Angle OAf because the lines AO and CRS are parallel. Now, it is found by observation, that the Angle OAL is much greater than the Angle OAf; but OAL is equal to ALC, and OAf is equal to ASC. Now, as ASC is much less than ALC, it proves that the Earth’s semi-diameter AC appears much greater as seen from the Moon at L than from the Sun at S: and therefore the Earth is much farther from the Sun than from the Moon[48]. The Quantities of these Angles are determined by observation in the following manner.
Let a graduated instrument as DAE, (the larger the better) having a moveable Index and Sight-holes, be fixed in such a manner, that it’s plane surface may be parallel to the Plan of the Equator, and it’s edge AD in the Meridian: so that when the Moon is in the Equinoctial, and on the Meridian at E, she may be seen through the sight-holes when the edge of the moveable index cuts the beginning of the divisions at o, on the graduated limb DE; and when she is so seen, let the precise time be noted. Now, as the Moon revolves about the Earth from the Meridian to the Meridian again in 24 hours 48 minutes, she will go a fourth part round it in a fourth part of that time, viz. in 6 hours 12 minutes, as seen from C, that is, from the Earth’s center or Pole. But as seen from A, the observer’s place on the Earth’s surface, the Moon will seem to have gone a quarter round the Earth when she comes to the sensible Horizon at O; for the Index through the sights of which she is then viewed will be at d, 90 degrees from D, where it was when she was seen at E. Now, let the exact moment when the Moon is seen at O (which will be when she is in or near the sensible Horizon) be carefully noted[49], that it may be known in what time she has gone from E to O; which time subtracted from 6 hours 12 minutes (the time of her going from E to L) leaves the time of her going from O to L, and affords an easy method for finding the Angle OAL (called the Moon’s horizontal Parallax, which is equal to the Angle ALC) by the following Analogy: As the time of the Moon’s describing the arc EO is to 90 degrees, so is 6 hours 12 minutes to the degrees of the Arc DdE, which measures the Angle EAL; from which subtract 90 degrees, and there remains the Angle OAL, equal to the Angle ALC, under which the Earth’s Semi-diameter AC is seen from the Moon. Now, since all the Angles of a right-lined Triangle are equal to 180 degrees, or to two right Angles, and the sides of a Triangle are always proportional to the Sines of the opposite Angles, say, by the Rule of Three, as the Sine of the Angle ALC at the Moon L is to it’s opposite side AC the Earth’s Semi-diameter, which is known to be 3985 miles, so is Radius, viz. the Sine of 90 degrees, or of the right Angle ACL to it’s opposite side AL, which is the Moon’s distance at L from the observer’s place at A on the Earth’s surface; or, so is the Sine of the Angle CAL to its opposite side CL, which is the Moon’s distance from the Earth’s centre, and comes out at a mean rate to be 240,000 miles. The Angle CAL is equal to what OAL wants of 90 degrees.
191. The Sun’s distance from the Earth is found the same way, but with much greater difficulty; because his horizontal Parallax, or the Angle OAS equal to the Angle ASC, is so small as, to be hardly perceptible, being only 10 seconds of a minute, or the 360th part of a degree. But the Moon’s horizontal Parallax, or Angle OAL equal to the Angle ALC, is very discernible; being 57ʹ 49ʺ, or 3469ʺ at it’s mean state; which is more than 340 times as great as the Sun’s: and therefore, the distances of the heavenly bodies being inversely as the Tangents of their horizontal Parallaxes, the Sun’s distance from the Earth is at least 340 times as great as the Moon’s; and is rather understated at 81 millions of miles, when the Moon’s distance is certainly known to be 240 thousand. But because, according to some Astronomers, the Sun’s horizontal Parallax is 11 seconds, and according to others only 10, the former Parallax making the Sun’s distance to be about 75,000,000 of miles, and the latter 82,000,000; we may take it for granted, that the Sun’s distance is not less than as deduced from the former, nor more than as shewn by the latter: and every one who is accustomed to make such observations, knows how hard it is, if not impossible, to avoid an error of a second; especially on account of the inconstancy of horizontal Refractions. And here, the error of one second, in so small an Angle, will make an error of 7 millions of miles in so great a distance as that of the Sun’s; and much more in the distances of the superiour Planets. But Dr. Halley has shewn us how the Sun’s distance from the Earth, and consequently the distances of all the Planets from the Sun, may be known to within a 500th part of the whole, by a Transit of Venus over the Sun’s Disc, which will happen on the 6th of June, in the year 1761; till which time we must content ourselves with allowing the Sun’s distance to be about 81 millions of miles, as commonly stated by Astronomers.
192. The Sun and Moon appear much about the same bulk: And every one who understands Geometry knows how their true bulks may be deduced from the apparent, when their real distances are known. Spheres are to one another as the Cubes of their Diameters; whence, if the Sun be 81 millions of miles from the Earth, to appear as big as the Moon, whose distance does not exceed 240 thousand miles, he must, in solid bulk, be 42 millions 875 thousand times as big as the Moon.
193. The horizontal Parallaxes are best observed at the Equator; 1. Because the heat is so nearly equal every day, that the Refractions are almost constantly the same. 2. Because the parallactic Angle is greater there as at A (the distance from thence to the Earth’s Axis being greater,) than upon any parallel of Latitude, as a or b.
194. The Earth’s distance from the Sun being determined, the distances of all the other Planets from him are easily found by the following analogy, their periods round him being ascertained by observation. As the square of the Earth’s period round the Sun is to the cube of it’s distance from him, so is the square of the period of any other Planet to the cube of it’s distance, in such parts or measures as the Earth’s distance was taken; see § 111. This proportion gives us the relative mean distances of the Planets from the Sun to the greatest degree of exactness; and they are as follows, having been deduced from their periodical times, according to the law just mentioned, which was discovered by Kepler and demonstrated by Sir Isaac Newton.
| Of Mercury | Venus | The Earth | Mars | Jupiter | Saturn |
|---|---|---|---|---|---|
| 87.9692 | 224.6176 | 365.2564 | 686.9785 | 4332.514 | 10759.275 |
| Relative mean distances from the Sun. | |||||
| 38710 | 72333 | 100000 | 152369 | 520096 | 954006 |
| From these numbers we deduce, that if the Sun’s horizontal Parallax be 10ʺ, the real mean distances of the Planets from the Sun in English miles are | |||||
| 31,742,200 | 59,313,060 | 82,000,000 | 124,942,580 | 426,478,720 | 782,284,920 |
| But if the Sun’s Parallax be 11ʺ their distances are no more than | |||||
| 29,032,500 | 54,238,570 | 75,000,000 | 114,276,750 | 390,034,500 | 715,504,500 |
| Errors in distance a rising from the mistake of 1ʺ in the Sun’s Parallax | |||||
| 2,709,700 | 5,074,490 | 7,000,000 | 10,665,830 | 36,444,220 | 66,780,420 |
195. These last numbers shew, that although we have the relative distances of the Planets from the Sun to the greatest nicety, yet the best observers have not hitherto been able to ascertain their true distances to within less than a twelfth part of what they really are. And therefore, we must wait with patience till the 6th of June, A. D. 1761; wishing that the Sky may then be clear to all places where there are good Astronomers and accurate instruments for observing the Transit of Venus over the Sun’s Disc at that time: as it will not happen again, so as to be visible in Europe, in less than 235 years after.
196. The Earth’s Axis produced to the Stars, being carried [50]parallel to itself during the Earth’s annual revolution, describes a circle in the Sphere of the fixed Stars equal to the Orbit of the Earth. But this Orbit, though very large in itself, if viewed from the Stars, would appear no bigger than a point; and consequently, the circle described in the Sphere of the Stars by the Axis of the Earth produced, if viewed from the Earth, must appear but as a point; that is, it’s diameter appears too little to be measured by observation: for Dr. Bradley has assured us, that if it had amounted to a single second, or two at most, he should have perceived it in the great number of observations he has made, especially upon γ Dragonis; and that it seemed to him very probable that the annual Parallax of this Star is not so great as a single second: and consequently, that it is above 400 thousand times farther from us than the Sun. Hence the celestial poles seem to continue in the same points of the Heavens throughout the year; which by no means disproves the Earth’s annual motion, but plainly proves the distance of the Stars to be exceeding great.
197. The small apparent motion of the Stars § 113, discovered by that great Astronomer, he found to be no ways owing to their annual Parallax (for it came out contrary thereto) but to the Aberration of their light, which can result from no known cause besides that of the Earth’s annual motion; and as it agrees so exactly therewith, it proves beyond dispute that the Earth has such a motion: for this Aberration compleats all it’s various Phenomena every year; and proves that the velocity of star-light is such as carries it through a space equal to the Sun’s distance from us in 8 minutes 13 seconds of time. Hence, the velocity of light is [51]10 thousand 210 times as great as the Earth’s velocity in it’s Orbit; which velocity (from what we know already of the Earth’s distance from the Sun) may be affected to be at least between 57 and 58 thousand miles every hour: and supposing it to be 58000, this number multiplied by the above 10210, gives 592 million 180 thousand miles for the hourly motion of light: which last number divided by 3600, the number of seconds in an hour, shews that light flies at the rate of more than 164 thousand miles every second of time, or swing of a common clock pendulum.